Skip to main content Accessibility help




We show how to use Jantzen’s sum formula for Weyl modules to prove semisimplicity criteria for endomorphism algebras of $\mathbf{U}_{q}$ -tilting modules (for any field $\mathbb{K}$ and any parameter $q\in \mathbb{K}-\{0,-1\}$ ). As an application, we recover the semisimplicity criteria for the Hecke algebras of types $\mathbf{A}$ and $\mathbf{B}$ , the walled Brauer algebras and the Brauer algebras from our more general approach.


Corresponding author


Hide All

Elements of the text in the article appear in colour online available at 10.1017/S1446788716000392.

H.H.A. was supported by the Center of Excellence grant ‘Centre for Quantum Geometry of Moduli Spaces (QGM)’ from the Danish National Research Foundation (DNRF), C.S. by a Hirzebruch professorship of the Max-Planck-Gesellschaft and D.T. by a research funding of the Deutsche Forschungsgemeinschaft (DFG) during this work.



Hide All
[1] Andersen, H. H., Tilting Modules for Algebraic and Quantum Groups, NATO Science Series II: Mathematics, Physics and Chemistry, 28 (Kluwer Academic, Dordrecht, 2001).
[2] Andersen, H. H. and Kulkarni, U., ‘Sum formulas for reductive algebraic groups’, Adv. Math. 217(1) (2008), 419447.
[3] Andersen, H. H., Polo, P. and Wen, K. X., ‘Representations of quantum algebras’, Invent. Math. 104(1) (1991), 159.
[4] Andersen, H. H., Stroppel, C. and Tubbenhauer, D., ‘Additional notes for the paper “Cellular structures using $\mathbf{U}_{q}$ -tilting modules”’, eprint,,,
[5] Andersen, H. H., Stroppel, C. and Tubbenhauer, D., ‘Cellular structures using $\mathbf{U}_{q}$ -tilting modules’, arXiv:1503.00224.
[6] Andersen, H. H. and Tubbenhauer, D., ‘Diagram categories for $\mathbf{U}_{q}$ -tilting modules at roots of unity’, Transform. Groups, to appear. Published online (25 January 2016).
[7] Ariki, S., ‘On the semi-simplicity of the Hecke algebra of (ℤ/rℤ) ≀ S n ’, J. Algebra 169(1) (1994), 216225.
[8] Brauer, R., ‘On algebras which are connected with the semisimple continuous groups’, Ann. of Math. (2) 38(4) (1937), 857872.
[9] Brown, W. P., ‘The semisimplicity of 𝜔 f n ’, Ann. of Math. (2) 63 (1956), 324335.
[10] Brundan, J. and Stroppel, C., ‘Gradings on walled Brauer algebras and Khovanov’s arc algebra’, Adv. Math. 231(2) (2012), 709773.
[11] Cox, A., De Visscher, M., Doty, S. and Martin, P., ‘On the blocks of the walled Brauer algebra’, J. Algebra 320(1) (2008), 169212.
[12] Dipper, R., Doty, S. and Stoll, F., ‘The quantized walled Brauer algebra and mixed tensor space’, Algebr. Represent. Theory 17(2) (2014), 675701.
[13] Dipper, R. and James, G., ‘Representations of Hecke algebras of type B n ’, J. Algebra 146(2) (1992), 454481.
[14] Donkin, S., The q-Schur Algebra, London Mathematical Society Lecture Note Series, 253 (Cambridge University Press, Cambridge, 1998).
[15] Donkin, S. and Tange, R., ‘The Brauer algebra and the symplectic Schur algebra’, Math. Z. 265(1) (2010), 187219.
[16] Doran, W. F. IV, Wales, D. B. and Hanlon, P. J., ‘On the semisimplicity of the Brauer centralizer algebras’, J. Algebra 211 (1999), 647685.
[17] Du, J., Parshall, B. and Scott, L., ‘Quantum Weyl reciprocity and tilting modules’, Comm. Math. Phys. 195(2) (1998), 321352.
[18] Ehrig, M. and Stroppel, C., ‘Schur–Weyl duality for the Brauer algebra and the ortho-symplectic Lie superalgebra’, Math. Z. 284(1–2) (2016), 595613.
[19] Goodman, R. and Wallach, N. R., Symmetry, Representations, and Invariants, Graduate Texts in Mathematics, 255 (Springer, Dordrecht, 2009).
[20] Graham, J. J. and Lehrer, G., ‘Cellular algebras’, Invent. Math. 123(1) (1996), 134.
[21] Gyoja, A. and Uno, K., ‘On the semisimplicity of Hecke algebras’, J. Math. Soc. Japan 41(1) (1989), 7579.
[22] Härterich, M., ‘Murphy bases of generalized Temperley–Lieb algebras’, Arch. Math. (Basel) 72(5) (1999), 337345.
[23] Hu, J., ‘Schur–Weyl reciprocity between quantum groups and Hecke algebras of type G (r, 1, n)’, Math. Z. 238(3) (2001), 505521.
[24] Hu, J., ‘BMW algebra, quantized coordinate algebra and type C Schur–Weyl duality’, Represent. Theory 15 (2011), 162.
[25] Hu, J. and Stoll, F., ‘On double centralizer properties between quantum groups and Ariki–Koike algebras’, J. Algebra 275(1) (2004), 397418.
[26] Humphreys, J. E., Representations of Semisimple Lie Algebras in the BGG Category 𝓞, Graduate Studies in Mathematics, 94 (American Mathematical Society, Providence, RI, 2008).
[27] Jantzen, J. C., ‘Darstellungen halbeinfacher algebraischer Gruppen und zugeordnete kontravariante Formen’, Bonn. Math. Schr. 67 (1973), 124 pages.
[28] Jantzen, J. C., ‘Darstellungen halbeinfacher Gruppen und kontravariante Formen’, J. reine angew. Math. 290 (1977), 117141.
[29] Jantzen, J. C., Lectures on Quantum Groups, Graduate Studies in Mathematics, 6 (American Mathematical Society, Providence, RI, 1996).
[30] Jantzen, J. C., Representations of Algebraic Groups, 2nd edn, Mathematical Surveys and Monographs, 107 (American Mathematical Society, Providence, RI, 2003).
[31] Kassel, C. and Turaev, V., Braid Groups, Graduate Texts in Mathematics, 247 (Springer, New York, 2008), with the graphical assistance of Olivier Dodane.
[32] Koike, K., ‘On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters’, Adv. Math. 74(1) (1989), 5786.
[33] Kuperberg, G., ‘Spiders for rank 2 Lie algebras’, Comm. Math. Phys. 180(1) (1996), 109151.
[34] Lehrer, G. and Zhang, R., ‘Strongly multiplicity free modules for Lie algebras and quantum groups’, J. Algebra 306(1) (2006), 138174.
[35] Lehrer, G. and Zhang, R., ‘The second fundamental theorem of invariant theory for the orthogonal group’, Ann. of Math. (2) 176(3) (2012), 20312054.
[36] Lusztig, G., Introduction to Quantum Groups, Modern Birkhäuser Classics (Birkhäuser/Springer, New York, 2010), reprint of the 1994 edition.
[37] Lyle, S. and Mathas, A., ‘Blocks of cyclotomic Hecke algebras’, Adv. Math. 216(2) (2007), 854878.
[38] Martin, P., ‘The structure of the partition algebras’, J. Algebra 183(2) (1996), 319358.
[39] Mathas, A., Iwahori–Hecke Algebras and Schur Algebras of the Symmetric Group, University Lecture Series, 15 (American Mathematical Society, Providence, RI, 1999).
[40] Mazorchuk, V. and Stroppel, C., ‘ G (, k, d)-modules via groupoids’, J. Algebraic Combin. 43(1) (2016), 1132.
[41] Morton, H. R., ‘A basis for the Birman–Wenzl algebra’, arXiv:1012.3116 (based on joint work with A. J. Wassermann).
[42] Năstăsescu, C., Raianu, Ş. and Van Oystaeyen, F., ‘Modules graded by G-sets’, Math. Z. 203(4) (1990), 605627.
[43] Paradowski, J., Filtrations of Modules Over the Quantum Algebra, Proceedings of Symposia in Pure Mathematics, 56 (American Mathematical Society, Providence, RI, 1994).
[44] Ram, A. and Ramagge, J., ‘Affine Hecke algebras, cyclotomic Hecke algebras and Clifford theory’, in: A Tribute to C. S. Seshadri, Trends in Mathematics (Birkhäuser, Basel, 2003), 428466.
[45] Rui, H., ‘A criterion on the semisimple Brauer algebras’, J. Combin. Theory Ser. A 111(1) (2005), 7888.
[46] Rui, H. and Si, M., ‘A criterion on the semisimple Brauer algebras. II’, J. Combin. Theory Ser. A 113(6) (2006), 11991203.
[47] Rui, H. and Si, M., ‘Gram determinants and semisimplicity criteria for Birman–Wenzl algebras’, J. reine angew. Math. 631 (2009), 153179.
[48] Ryom-Hansen, S., ‘A q-analogue of Kempf’s vanishing theorem’, Mosc. Math. J. 3(1) (2003), 173187; 260.
[49] Sakamoto, M. and Shoji, T., ‘Schur–Weyl reciprocity for Ariki–Koike algebras’, J. Algebra 221(1) (1999), 293314.
[50] Sawin, S. F., ‘Quantum groups at roots of unity and modularity’, J. Knot Theory Ramifications 15(10) (2006), 12451277.
[51] Schoutens, H., The Use of Ultraproducts in Commutative Algebra, Lecture Notes in Mathematics, 1999 (Springer, Berlin, 2010).
[52] Thams, L., ‘Two classical results in the quantum mixed case’, J. reine angew. Math. 436 (1993), 129153.
[53] Turaev, V. G., ‘Operator invariants of tangles, and R-matrices’, Izv. Akad. Nauk SSSR Ser. Mat. 53(5) (1989), 10731107; 1135. Translation in Math. USSR-Izv. 35(2) (1990), 411–444.
[54] Wenzl, H., ‘On the structure of Brauer’s centralizer algebras’, Ann. of Math. (2) 128(1) (1988), 173193.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Related content

Powered by UNSILO




Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.